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Approximate Distance Queries and Compact Routing in Sparse Graphs
"... Abstract—An approximate distance query data structure is a compact representation of a graph, and can be queried to approximate shortest paths between any pair of vertices. Any such data structure that retrieves stretch 2k−1 paths must require spaceΩ(n 1+1/k) for graphs of n nodes. The hard cases th ..."
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Abstract—An approximate distance query data structure is a compact representation of a graph, and can be queried to approximate shortest paths between any pair of vertices. Any such data structure that retrieves stretch 2k−1 paths must require spaceΩ(n 1+1/k) for graphs of n nodes. The hard cases that enforce this lower bound are, however, rather dense graphs with average degreeΩ(n 1/k). We present data structures that, for sparse graphs, substantially break that lower bound barrier at the expense of higher query time. For instance, general graphs require O(n 3/2) space and constant query time for stretch 3 paths. For the realistic scenario of a graph with average degreeΘ(log n), special cases of our data structures retrieve stretch 2 paths with Õ(n 3/2) space and stretch 3 paths with Õ(n) space, albeit at the cost of Õ ( � n) query time. Moreover, supported by largescale simulations on graphs including the ASlevel Internet graph, we argue that our stretch2 scheme would be simple and efficient to implement as a distributed compact routing protocol. I.
Scalable routing easy as pie: A practical isometric embedding protocol
 in Network Protocols (ICNP), 2011 19th IEEE International Conference on, 2011
"... We present PIE, a scalable routing scheme that achieves 100 % packet delivery and low path stretch. It is easy to implement in a distributed fashion and works well when costs are associated to links. Scalability is achieved by using virtual coordinates in a space of concise dimensionality, which ena ..."
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We present PIE, a scalable routing scheme that achieves 100 % packet delivery and low path stretch. It is easy to implement in a distributed fashion and works well when costs are associated to links. Scalability is achieved by using virtual coordinates in a space of concise dimensionality, which enables greedy routing based only on local knowledge. PIE is a general routing scheme, meaning that it works on any graph. We focus however on the Internet, where routing scalability is an urgent concern. We show analytically and by using simulation that the scheme scales extremely well on Internetlike graphs. In addition, its geometric nature allows it to react efficiently to topological changes or failures by finding new paths in the network at no cost, yielding better delivery ratios than standard algorithms. The proposed routing scheme needs an amount of memory polylogarithmic in the size of the network and requires only local communication between the nodes. Although each node constructs its coordinates and routes packets locally, the path stretch remains extremely low, even lower than for centralized or less scalable stateoftheart algorithms: PIE always finds short paths and often enough finds the shortest paths. Abstract — 1 I.
Sparse spanners vs. compact routing
 IN: SPAA
, 2011
"... Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables of ˜ Θ (√n) bits 1 per node. The space lower bound is due to the existence of dense graphs with large girth. Dense graphs ..."
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Cited by 5 (2 self)
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Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables of ˜ Θ (√n) bits 1 per node. The space lower bound is due to the existence of dense graphs with large girth. Dense graphs can be sparsified to subgraphs, called spanners, with various stretch guarantees. There are spanners with additive stretch guarantees (some even have constant additive stretch) but only very few additive routing schemes are known. In this paper, we give reasons why routing in unweighted graphs with additive stretch is difficult in the form of space lower bounds for general graphs and for planar graphs. We prove that any routing scheme using routing tables of size µ bits per node and addresses of polylogarithmic length has additive stretch ˜ Ω ( p n/µ) for general graphs, and ˜ Ω ( √ n/µ) for planar graphs, respectively. Routing with tables of size Õ(n1/3) thus requires a polynomial additive stretch of ˜Ω(n 1/3), whereas spanners with average degree O(n 1/3) and constant additive stretch exist for all graphs. Spanners, however sparse they are, do not tell us how to route. These bounds provide the first separation of sparse spanner problems and compact routing problems. On the positive side, we give an almost tight upper bound: we present the first nontrivial compact routing scheme with o(lg 2 n)bit addresses, additive stretch Õ(n1/3), and table size Õ(n1/3) bits for all graphs with linear local treewidth such as planar, boundedgenus, and apexminorfree graphs.
ShortestPath Queries for Complex Networks: Exploiting Low Treewidth Outside the Core
"... We present new and improved methods for efficient shortestpath query processing. Our methods are tailored to work for two specific classes of graphs: graphs with small treewidth and complex networks. Seemingly unrelated at first glance, these two classes of graphs have some commonalities: complex ne ..."
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We present new and improved methods for efficient shortestpath query processing. Our methods are tailored to work for two specific classes of graphs: graphs with small treewidth and complex networks. Seemingly unrelated at first glance, these two classes of graphs have some commonalities: complex networks are known to have a core–fringe structure with a dense core and a treelike fringe. Our main contributions are efficient algorithms and data structures on three different levels. First, we provide two new methods for graphs with small but not necessarily constant treewidth. Our methods achieve new tradeoffs between space and query time. Second, we present an improved treedecompositionbased method for complex networks, utilizing the methods for graphs with small treewidth. Third, we extend our method to handle the highly interconnected core with existing exact and approximate methods. We evaluate our algorithms both analytically and experimentally. We prove that our algorithms for lowtreewidth graphs achieve improved tradeoffs between space and query time. Our experiments on several realworld complex networks further confirm the efficiency of our methods: Both the exact and the hybrid method have faster preprocessing and query times than existing methods. The hybrid method in particular provides an improved tradeoff between space and accuracy.
Evaluating Compact Routing Algorithms on RealWorld Networks
, 2010
"... Compact routing has shown promise for reducing the forwarding state in Internetlike graphs, without badly impacting traffic flows. This dissertation compares two such compact routing algorithms on real Internet snapshots from router data across 12 years. The results indicate that these algorithms be ..."
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Compact routing has shown promise for reducing the forwarding state in Internetlike graphs, without badly impacting traffic flows. This dissertation compares two such compact routing algorithms on real Internet snapshots from router data across 12 years. The results indicate that these algorithms behave consistently over time, and exhibit extremely small forwarding tables with very low path inflation. Acknowledgements Colin Perkins and Stephen Strowes for their support, feedback, discussions, helpful pointers, field trips, anecdotes, and, of course, the initial idea for this project. The departmental support staff for their help through out the year with machine upkeep, and allowing access to copious amounts of server space, hard drives, and docks; their assistance was invaluable. The Embedded, Networked and Distributed Systems group for their help and support through out. The Algorithms group for sanity checking any graph theory and algorithms I produced. My fellow MSci and MRes students for their company, help, and generally making this year bearable. Finally, the Level 4 and 3 students for their patience while I stole their CPU cycles, RAM modules, and hard drive space. ii 5.2.3 Routing............................. 29 5.3 Summary................................ 29 6 BradyCowen (BC) compact routing 30
Toward a Distance Oracle for BillionNode Graphs
, 2013
"... The emergence of real life graphs with billions of nodes poses significant challenges for managing and querying these graphs. One of the fundamental queries submitted to graphs is the shortest distance query. Online BFS (breadthfirst search) and offline precomputing pairwise shortest distances are ..."
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The emergence of real life graphs with billions of nodes poses significant challenges for managing and querying these graphs. One of the fundamental queries submitted to graphs is the shortest distance query. Online BFS (breadthfirst search) and offline precomputing pairwise shortest distances are prohibitive in time or space complexity for billionnode graphs. In this paper, we study the feasibility of building distance oracles for billionnode graphs. A distance oracle provides approximate answers to shortest distance queries by using a precomputed data structure for the graph. Sketchbased distance oracles are good candidates because they assign each vertex a sketch of bounded size, which means they have linear space complexity. However, stateoftheart sketchbased distance oracles lack efficiency or accuracy when dealing with big graphs. In this paper, we address the scalability and accuracy issues by focusing on optimizing the three key factors that affect the performance of distance oracles: landmark selection, distributed BFS, and answer generation. We conduct extensive experiments on both real networks and synthetic networks to show that we can build distance oracles of affordable cost and efficiently answer shortest distance queries even for billionnode graphs.
Brief Announcement: Revisiting the Powerlaw Degree Distribution for Social Graph Analysis ABSTRACT
"... {alessandra, htzheng, ravenben} ..."
Compact Routing for the Future Internet
, 2012
"... The Internet relies on its interdomain routing system to allow data transfer between any two endpoints regardless of where they are located. This routing system currently uses a shortest path routing algorithm (modified by local policy constraints) called the Border Gateway Protocol. The massive gr ..."
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The Internet relies on its interdomain routing system to allow data transfer between any two endpoints regardless of where they are located. This routing system currently uses a shortest path routing algorithm (modified by local policy constraints) called the Border Gateway Protocol. The massive growth of the Internet has led to large routing tables that will continue to grow. This will present a serious engineering challenge for router designers in the longterm, rendering state (routing table) growth at this pace unsustainable. There are various shortterm engineering solutions that may slow the growth of the interdomain routing tables, at the expense of increasing the complexity of the network. In addition, some of these require manual configuration, or introduce additional points of failure within the network. These solutions may give an incremental, constant factor, improvement. However, we know from previous work that all shortest path routing algorithms require forwarding state that grows linearly with the size of the network in the worst case. Rather than attempt to sustain interdomain routing through a shortest path routing algorithm, compact routing algorithms exist that guarantee worstcase sublinear state requirements at all nodes by allowing an upperbound on path length relative to the theoretical shortest path,